Revista Electrónica Nova ScientiaHernández-González, S. et al. Revista Electrónica Nova Scientia Numerical analysis of minimum cost network flow with queuing stations: the M/M/1

Hernández-González, S. et al.

Revista Electrónica Nova Scientia

Numerical analysis of minimum cost network

flow with queuing stations: the M/M/1 case

Análisis numérico de flujo a costo mínimo en

redes con estaciones de espera: el caso M/M/1

Salvador Hernández-González1, Idalia Flores de la Mota2,

José Alfredo Jiménez-García1 y Manuel Darío Hernández-

Ripalda1

1Departamento de Ingeniería Industrial, Instituto Tecnológico de Celaya 2 Facultad de Ingeniería, Universidad Nacional Autónoma de México

México

Salvador Hernández González. E-mail: [email protected]

© Universidad De La Salle Bajío (México)

mailto:[email protected]


Nova Scientia ISSN 2007 - 0705, Nº 18 Vol. 9 (1), 2017. pp: 257 – 289 - 258 -

Resumen

En una red los nodos representan estaciones, almacenes, centros de distribución y clientes y

circulan tanto materiales como información, por lo que una herramienta utilizada para apoyar la

toma de decisiones es el modelo de flujo a costo mínimo que toma en cuenta únicamente costos

de transporte. En la realidad los nodos prestan un servicio el cual requiere un tiempo de servicio,

la atención sigue una disciplina y además se forma una fila de espera. En este trabajo se propone

una modificación del modelo de flujo a costo mínimo para la optimización en redes de líneas de

espera. Se resolvieron varios casos donde se observa un grado de exactitud aceptable en el

cálculo del tiempo de ciclo y trabajo en proceso. El trabajo es de interés para los administradores

y/o responsables de las cadenas de abasto y útil para la toma de decisiones a mediano y largo

plazo.

Palabras clave: flujo en redes; cadena de abasto; redes de colas; tiempo de ciclo; trabajo en

proceso

Recepción: 19-01-17 Aceptación: 05-03-17

Abstract In a network the nodes represent stations, warehouses, distribution centers and customers and not

just materials but also information circulate, so the minimum cost flow model, that only takes

transport costs into account is one of the tools used to support the decision-making. In reality, the

nodes provide a service which requires a service time, the servicing follows a discipline and also

a queue is formed, generating the respective service and queuing costs. A modified version of the

minimum cost flow model is proposed in this paper for the optimization of the flow in a queuing

network. A variety of cases were solved. An acceptable level of accuracy was observed in the

calculation of the time cycle and work in progress. The results indicate that the optimum solution

tends to balance the workload along the network. This paper is of interest to administrators and

people in charge of supply chains and is useful for decision-making in the medium or long time.

Keywords: network flow, supply chain, queueing networks, cycle time, work in process

Numerical analysis of minimum cost network flow with queuing stations: the M/M/1 case

Nova Scientia ISNN 2007 - 0705, Nº 18 Vol. 9 (1), 2017. pp: 257 – 289 - 259 -

Introduction

In industry and services, supply systems have a high degree of complexity, owing to the existence

of the multiple relations that each element has with others within the same system. In

consequence, it is often observed that work should be done with a network of stations/plants,

warehouses/suppliers, containers/distribution centers and customers. Throughout such a network

not only materials but also information circulates (Bhaskar and Lallement, 2010).

Some examples of the decisions that the administrators of these systems have to take are: how

much product should be manufactured or how much should be sent from each workplace to the

following link in the chain, with the total cost of dispatch being the common performance

measurement (Figure 1). Consider also a cross-docking system where goods and materials are

transferred without actually entering the warehouse or being put away into storage (Kulwiec,

2004); a manager must determine the service time or the number of servers that reduces the

residence time within the facilities, in other words the cycle time in the system.

Plant 1

Plant 2

Plant 3

Supplier 1

Supplier 2

Supplier 3

Distributor 1

Distributor 2

Distributor 3

Customer

1

Customer

2

Customer

3

Suppliers Manufacturing Distribution Customers

Figure 1. Network of factories, warehouses and customers

Optimal network design plays an important role in the supply chain operation, as good logistic

distribution network can save transportation costs as well as improve service levels (Marmolejo,

Rodríguez, Cruz-Mejía & Saucedo, 2016). The minimum cost network flow model is a useful

tool for analyzing these types of systems.



It is worth mentioning that the nodes in a network, such as the one mentioned above are usually

stations that provide a service, in other words, we are talking about queuing networks. Analytical

queuing models are used for determining properties such as work in progress, time in the system

(cycle time), congestion of the system, measuring the quality of the service as well as quantifying

the operating cost of said systems.

Networks may be open (the customers can enter and leave through any station on the network) or

closed (the customer circulates around the network). Queuing network models are employed to

represent not only production or manufacturing lines or, but also service and supply chain

systems (Bolch, Greiner, de Meer and Trivedi, 2006).

Problem

The minimum cost network flow model only takes the transport costs into account; however

every node on the network is often a station that provides a service (a factory, a warehouse). This

operation generates a service cost (the products have to be manufactured) and a cost for the time

taken to be served (the orders follow a service discipline and a queue is formed). In this context,

managers and practitioners also need information about cycle time, work-in-process as well as

the availability of resources at plants and warehouses. For a given topology of plants, warehouses

and demand centers, it is evident that the solution obtained by a minimum cost flow model will

change if each node is considered as a queueing system but, how much is the difference?; how

sensitive is the solution if practitioners include transportation cost, waiting cost and service cost;

what a practitioner and a manager can expect if they wish to measure supply chain performance

as queueing networks.

Background

Most papers on queuing networks focus on the determination of the properties of the networks

and use both analytical means and simulation. These models can be used for finding properties,

control, planning and optimization, which is the case with this paper.

In the paper of Pourbabai, Blanc and van der Duyn Schouten (1996) a modified version is

presented of the minimum cost flow model together with the respective version of the maximum

network flow problem with stations where queues form. Their paper proposes the probability of

meeting service levels as a constraint.



Kerbache and MacGregor Smith (2000) propose a mathematical programming model that

determines the optimum path that maximizes the throughput, the properties are calculated by

applying the Generalized Expansion Method. Given that this is a mixed integer nonlinear model,

an algorithm is developed to find good solutions, Yenisey (2006) applies concepts from

Pourbabai et al (1996) to construct a mathematical programming model for MRP systems.

MacGregor Smith (2011) considers M/G/c, M/G/c/c systems and proposes a procedure for

finding the optimum flow paths in a queuing network: In this case, the objective function is the

system’s throughput and is applied to the modeling of passenger flow in underground transport

systems.

Morabito, de Souza and Vázquez (2014) propose an analytical method for the shortest path

problem in G/G/c queuing networks as well as the application of equations to improve the

accuracy of the calculations of the system’s properties, while, when applicable, the objective

function also maximizes the throughput of the system. van Woensel T., Cruz F.R.B. (2014)

extended the model to G/G/c/K networks to determine the shortest route in systems with multiple

servers.

Some examples of applied cases we can mention are Bhaskar and Lallement’s paper (2010) for

the analysis of the supply chain of an Indian textile company, Srivathsan and Kamath (2012);

they propose a model for the analysis of supply chains, but raise the idea of using the model to

plan and find the optimum values of decision variables. He and Hu (2014) propose a model for

decision-making in disaster prevention systems for the Shanghai area, while Beard and

Chamberlain (2013) employ the principles proposed in Pourbabai. et al (1996) for the analysis of

the flow of information in computer network systems.

Other analytical techniques for the modelling of supply chains are Petri Nets, Series parallel

graphs, Markov chains and system dynamics (Srinivasa &Viswanadham, 2001).

This paper’s contributions are:

1. This paper is a variant on the proposal of Pourbabai, et al (1996), but in this case the queuing

and node operating costs are included in the objective function.

2. It differs from Morabito, Souza and Vázquez’s paper (2014) in that different versions of the

cost model are employed and assessed; the analysis is, for the moment, confined to the fact that

the service time and demand follow an exponential distribution.



3. We apply the decomposition principle to obtain the performance measures. MacGregor Smith

(2011), Morabito, et al.(2014), or van Woensel T., Cruz F.R.B. (2014) use General Expansion

Method.

4. We are contributing with an approach for the process of modeling and analysis of a supply

chain as a network of queues.

5. We obtain an optimal solution and calculate the work in process and cycle time of proeuction

orders.

6. Numerical examples are solved and the results are compared with those obtained from a

conventional minimum cost flow model. Simulation is used to validate analytical results of cycle

time and work-in-process.

This paper is of interest to administrators and/or people in charge of supply chains, purchasing

schedulers or production schedulers and is useful for decision-making in the medium or long

time. With this model we get information about the cycle time, the work in process and the

congestion at each station as well as the quantity of a product that must be sent to each customer

in the network, minimizing the total operating cost of the system, this approach takes ino account

the uncertainty of demand and service time of plants and distributors.

The minimum cost network flow problem

Consider a directed network that consists of a finite set of nodes N and a set of directed arcs A

that join pairs of N nodes. Arc (i, j) is said to be incident to nodes i and j and is directed from

node i to node j. Every node i is assigned a number fi that represents the available supply of an

article (f > 0) or the demand for some article (f < 0). Nodes with supply are known as source

nodes and the demand nodes are known as destination nodes. If fi= 0 then it is a transfer node,

given that there is no supply or demand at said node.

Each arc is associated with a value xij that is the flow of material from node i to node j and a

constant cij that is the cost of circulating through said arc. We shall that the total supply and the

total demand are equal, in other words 0 if . The minimum cost network flow problem can

be posed as follows: Shipping the available supply through the network in order to satisfy the

demand at a minimum cost. Mathematically, this problem is expressed as follows:

Minimize:



n

i

m

j

ijijxc1 1

(1)

Subject to:

i

m

k

ki

m

i

ij fxx 11

, ni ,...,2,1 (2)

0ijx (3)

The constraints (2) are called network flow conservation equations. In conservation equations,

the term

m

i ijx1

represents the total flow that leaves node i towards any node j, while the term

m

k kix1

is the flow that enters node i from any other node k.

The equations require the net flow that leaves node i to be equal to fi. Finally, it must be pointed

out that there can be flow capacity through the arcs.

The minimum cost flow model can be used to represent a logistics network where people and/or

materials move through various points, the movement of locomotives, communications systems,

oil pipelines, tanker schedules, etc (Bazaraa, Jarvis and Sherali, 1974, Wagner, 1975).

The minimum cost network flow problem can be posed as a linear programming problem and the

simplex method can be applied to find the optimum solution for a particular instance.

One of the assumptions in the aforementioned flow model is that no operation is done in the

nodes and that the material flows without undergoing any kind of delay. However the people in

charge of the administration and planning of the operations must take into account the time

customers and materials spend in a queue at the station while waiting for the corresponding

service; the time required to process a product, order or job; and the cost of the service, while

they must also determine and analyze the system’s properties such as the amount of work in

process (WIP) and the cycle time in the system (from the time when the order is received until it

is delivered to the end customer).

The aforementioned properties are performance measurements for queuing networks. A summary

of the method for calculating said properties is given below.



Queuing networks

A queuing network is one where there are at least two stations connected to each other (Figure 2).

Each node on the network represents a resource (for example a station or a factory). In this paper

we use the following premises:

1. The stations have infinite capacity in the line.

2. Customers arrive at the network at the rate of λ customers per unit of time and travel from one

station to another within the network.

3. Each station has si servers with a capacity of µi customers per unit of time.

1

3

2

CT3

WIP3

CT2

WIP2

CT1

WIP1

λ1

λ2

λ3

Figure 2. Queuing network

Properties of queuing networks.

Assume a network of n stations, the transit of the customers from one station to another is defined

by the transit matrix P where each element pij is the probability that the customer goes from

station i to station j. P is a square matrix. Assume that, through node i, γ0i customers per unit of

time can enter the network from the exterior. Therefore the total flow to any node i on the

network satisfies the following expression (Buzacott and Shanthikumar, 1993, Curry and

Feldman, 2011, Bolch et al, 2006):

m

j

ji

T

ijii p1

0 mi ,,2,1 (4)

Expressing this result in a matrix form:

λPγλT

(5)



Where γ and λ are the flow column vectors from the exterior to each node and the entry flow to

the node, respectively and PT is the transpose of the probability matrix. The system can be solved

to obtain the entry flows to each station λ:

γPIλ1

T (6)

Where I is the identity matrix. As can be observed, given γ and P, the equation calculates the

entry flows to each station on the network.

Quadratic coefficients of the entry flows

After the equation (4), the entry flow to each station is made up by two elements: the entry from

outside to the system through node i and the flow from any station within the network.

Assume that the external flow to a node j has a coefficient of variation that is denoted as jCa ,02 ,

likewise, the coefficient of variation for the flow from a node k to node j is denoted as jkCa ,2 .

If the station has s servers, and each customer requires 1 units of time to be served, with a

coefficient of variation jCST

2, also if each station that makes up the network is stable, in other

words it holds that 1, j

jjS

s

T (the used capacity or congestion of the work station) therefore

the quadratic coefficients of variation in the entry flows to each station are obtained with the

following approximation:

m

k kj

k

kT

kkj

ajkj

j

kjK

a

j

j

ap

s

skCp

kCpp

jCjCS

1

2

2

22

202

11

1

,0

(7)

Once again this approximation can be expressed in matrix form and is solved as a system of

linear equations, whose solution shall be:

βQIC2

a

1 (8)

Where each element of the matrix Q is calculated as follows:

j

kkjk

kj

pq

22 1 (9)

And each element of the vector β is obtained with:



m

k

kj

k

kT

kkj

j

kjK

a

j

j

ai ps

skCp

pjCjC S

1

2

2202 11

,0

(10)

Once the flows and their respective coefficient of variation have been obtained, the next step will

consist of assessing the properties of each one of the network’s station applying the

decomposition principle. The expressions for the cycle time of station i and variability of

departure from station i are:

iSiS

i

iiSid

i TTCC

CT ,,

2

,

2

1,

12

(11)

2

,

22

1,

2

, 1 iSiidiid CCC (12)

A summary of the procedure is given in figure 3 (Buzacott and Shanthikumar, 1993, Curry and

Feldman, 2011).

1. Read the vector of service times tS, the vector of coefficient of variation

Cts2, the vector of entry flows to each station γ and the vector of

coefficients of variation Ca2 at the entry.

2. Read probability matrix P

3. Calculate the vector of entry flows λ at each station

4. Assess ρ for each station, if ρ > 1 for any station, stop, the network is

unstable. Otherwise go to step 5.

5. Calculate the vector of quadratic coefficients of variation Ca2 at the

entry.

6. Calculate the properties of each station: Work in process, cycle time.

Figure 3. Procedure for calculating the properties of a queuing network

Cost function in a queuing network

The operating cost of a queuing network is expressed by the following equation:

FlowCostWIPCostServerCostOCOpCost (13)

m

i

m

j

ijij

m

i

iiW IP

m

i iS

iiS xcWIPC

T

sCOC

1 11

,

1 ,

, (14)



Conventionally, only the service cost and queuing cost are included in the operating cost (OC) in

a queuing network. Likewise, only the network flow cost is included in a minimum cost flow

problem.

Equation (13) includes the three terms: the cost arising from the operation of the servers

(

iS

iiS T

sC

,, ), the cost of having inventory in process in the station ( iiWIP WIPC ,

) and

finally the cost of transiting from station i to station j ( ijij xc ,).

As can be appreciated, flow xij between pairs of nodes is calculated using (4). Assuming that the

external flows only appear at the supply nodes, then the term γ0i is zero, and the flow for any

node other than the supply nodes is:

T

ijjij px

(15)

Optimization model

Assume that we have a network with a set of nodes Nn and set of arcs Aa . There is a set of

nodes that represent the customers and generate a demand of i units per unit of time; said

demand follows a probability distribution G.

The material receives a transformation at each node that, on average, requires iST ,units of time

and follows a probability distribution G. Therefore this is a network with G/G/s type stations. If

12 STC and 12 aC , then the station is Markovian or M/M/s, in accordance with the notation of

Kendall.

There is no external flow to the demand nodes. Moreover, reflows are only considered for

representing reprocesses, whereas exits from the system through any node other than the demand

nodes are not considered.

The flow between the network’s stations can be represented as a probability matrix P.

There is a service cost, Service Costi, for serving or producing a product unit, and there is also a

cost per unit of work in process, Work in process Costi. Finally there is also a cost for transiting

from one station to another cij. The proposed optimization model is as follows:

Minimize:

m

i

m

j

ijij

m

i

iiWIP

m

i iS

iiS xcWIPC

T

sCOC

1 11

,

1 ,

, (16)



Subject to:

i

m

k

ki

m

i

ij fxx 11

, ni ,,2,1 (17)

supinf

iii (18)

supinf

ijijij ppp (19)

11

m

i

ijp (20)

m

k

k

T

kiiij px1

0 (21)

The quantity of material that should circulate between pairs of nodes or stations should be

determined in such a way as to minimize the value of operation cost (16). It is worth mentioning

that in the model, the decision variable is the probability or fraction ijp of material that travels

through the arc, with which the flow through the arc is calculated afterwards.

The flow is constrained to the entry – throughput balance at each node (17) being satisfied, as

well as to the flow in the system being such that the congestion or used capacity at each station ρ

(used capacity or congestion) are fulfilled or, if not, to ρ < 1 for all the stations (18).

Likewise, the way in which the flow is directed through the network is constrained to the range

given in (19). In the model, the constraint (20) ensures that the material does not leave the

network through any node that is not a demand node. In any event this constraint can be modified

if necessary. The flows are calculated using (21).

The problem of routing customers, packets and other entity flows in finite queueing networks is

an integer nonlinear programming. Our model is a variant of the Optimal Routing Problem in

queueing networks which is known to be NP-hard (MacGregor Smith and Daskalaki, 1988,

Kerbache and MacGregor Smith, 2000). The proposed model can incorporate constraints for the

work in process or the cycle time. It is also feasible that the service time of one or more stations

and the number of servers are also decision variables.

It is worth mentioning that, like any model, it is necessary to be careful to take the necessary

factors into account and thus avoid situations such as getting a very complex model that does not

have a feasible solution.



This model can be applied to the analysis and solution of problems involved with the planning of

supply chain operations, using a networks approach and properties, such as the cycle time, work

in process, the load at the stations or production or processing centers.

Implementation on a spreadsheet

It was necessary, for the study, to construct a spreadsheet template using the steps of the

aforementioned figure 3.

The template automatically calculates the total entry flows of each node using the equations (4) –

(6); the flows through each arc with their respective cost; the costs of the stations; cycle time and

amount variability of departures of each station; as well as the total operating cost of the system

using the equations (7) – (12).

Work in process of each station is obtained with Little’s Law:

CTWIP (22)

The template allows us to model systems that have up to 10 stations with a network arrangement

or else with a pure series arrangement. Further details will be encountered in Appendix 1.

Numerical examples

A numerical analysis was carried out of several cases with a proposed minimum cost flow model,

estimating properties (Work in process and cycle time), while the results were validated using

simulation.

All cases were built taking into account that the maximum number of stages that are considered

in mathematical models of supply chains are four (Olhager, Pashaei and Sternberg, 2015). Case

A is a conventional minimum cost flow model without operation at the nodes. In series B, the

minimum cost flow model with operation at the nodes was studied with and without costs (Table

1).

In case B1, decision variables are the probabilities pij of the customer going from node i to node j.

In case B2, the decision variables are pij as well as the service time at each station or node. In

case B3, which is the model with service cost and queuing cost, the decision variable is the

fraction pij that travels between pairs of nodes.



Table 1. Cases studied

Case

Operation

at the

nodes

Cost

flow

Cost of

service

and

queuing

Decision variable

A No Yes No Flow at each arc xij

B1 Yes Yes No

Fraction pij that is sent from node i

to node j. The flow is calculated

using (13).

B2 Yes Yes No


to node j and service time at each

node. The flow is calculated using

(13).

B3 Yes Yes Yes


to node j. The flow is calculated

using (13).

B4 Yes Yes Yes


to node j and service time at each

node. The flow is calculated using

(13).

Finally, in model B4, the decision variables are the fraction pij that travels between pairs of nodes

and the service time at each station or node.

To validate each analytical result from series B (flows, WIP and cycle time), we used a

simulation model that was constructed in the Arena simulation software. Five independent

replications were done. The run length was chosen to obtain a 95% confidence interval of WIP

and cycle time. The details of the model and the data for the repeated samples are given in table

2. An example is given of the model in figure 4 as an illustration.



Supplier 1

Supplier 2

Manufacturer

1

Manufacturer

2

Distributor 1

Distributor 2

Manufacturer

3

Customer

2

Customer

1

Customer

3

Suppliers Manufacturing Distributors Customers

Order

Order

Figure4. Example of the simulation model for validating the results of series B

Table 2. Characteristics of the simulation model

Simulation time (95% confidence) 5000 days

Replications 5 per series B case

Calculated properties Work in process and cycle time

Probability function for service

time

Exponential

Probability function for demand Exponential

Series A

Consider a system such as that of figure 5. Nodes 1 and 2 are the sources of the supply, nodes 3

and 4 are manufacturing nodes, 5, 6 and 7 are transfer nodes and nodes 8 and 9 are the demand

nodes.



1

2

3

4

5

6

8

9

7


Figure 5. Directed network for case A

The costs per unit transported between pairs of nodes are given in table 3. The demand at nodes 8

and 9 is 30 units, while the supply available at nodes 1 and 2 is 30 units.

Table 3. Origin – destination costs

1 2 3 4 5 6 7 8 9

1 $ 25.0 $50.0

2 $33.0 $29.0

3 $22.0 $17.0 $21.0

4 $20.0 $20.0 $19.0

5 $20.0 $12.0

6 $14.0 $13.0

7 $12.0 $11.0

On solving the minimum cost flow model, using Solver and the data, we get the solution given in

figure 6: 30 units should be sent from node 1 to node 3, and 30 units from node 2 to node 4. 30

units shall be sent from node 3 to node 5, 30 units shall be sent from node 4 to node 7.



1

2

3

4

5

6

8

9

7


Figure 6. Solution to case A

Finally node 5 shall send 30 units to demand node 9 and node 7 shall supply 30 units to demand

node 8 (Table 4). The operating cost is $3570.

Table 4. Flow per arc

Arc 1 - 3 1- 4 2- 3 2 - 4 3- 5 3 - 6 3 - 7 4 - 5

Flow 30 0 0 30 30 0 0 30

Arc 4 - 6 4 - 7 5 - 8 5 - 9 6 - 8 6 - 9 7 - 8 7 - 9

Flow 0 0 0 30 0 0 30 0

Series B. Minimum cost flow model with operation at the nodes

The analysis of the system is presented below, incorporating the data about the operation at each

node, as well as the constraints on flow and use of the capacity.

Case B1

Consider the above minimum cost flow problem, taking into account that the nodes are stations

where the customers receive a service, the data corresponding to the service time at each station

are given in table 5.



Table 5. Service time and standard deviation of the service time at each node

Node 1 2 3 4 5 6 7

TS 0.01 0.028 0.027 0.03 0.06 0.03 0.03

σTs 0.01 0.028 0.027 0.03 0.06 0.03 0.03

We will assume, in this case, that the service times are exponentially distributed, there are only

transport costs between stations and that there is only one server at each node.

The decision variable is fraction pij which must be sent from station i to station j, this value is

used to calculate amount of the flow of material between each station (Table 6).

Table 6. Constraints of case B1

90.025.0 i , 7,6,5,4,3,2,1i Used capacity constraint

125.0 13 p , 125.0 14 p

125.0 23 p , 125.0 24 p

125.0 35 p , 125.0 36 p , 125.0 37 p

125.0 45 p , 125.0 46 p , 125.0 47 p

125.0 58 p , 125.0 59 p ,

125.0 68 p , 125.0 69 p ,

125.0 78 p , 125.0 79 p ,

Constraints for the

proportion that has to

move through arc ij

30787686585 ppp

30797696595 ppp

Node 8 demand

constraint

Node 9 demand

constraint

30

30

242232

141131

pp

pp

0

0

242141474464454

232131373363353

ppppp

ppppp

Constraint for supply

nodes 1 and 2

Node 3 flow constraint.

Node 4 flow constraint.

The constraints for the value of pij are given in table 6. For this example in particular, the used

capacity is constrained to the (25% – 90%) range.

In this case we assume that the total demand of nodes 8 and 9 is shared out equally between

nodes 1 and 2 (this could be the result of a decision taken beforehand by the person in charge in



accordance with some prior criterion) in consequence, the entry flow to said stations is 30 each.

The same table 6 shows the flow constraints for the supply nodes 1 and 2, as well as the demand

nodes 8 and 9. Of transfer nodes 3 – 7, only the equations corresponding to nodes 3 and 4 are

shown.

The flows between stations were obtained by running the Solver complement and are given in

table 7.

Table 7. Flow per arc

Arc 1 – 3 1- 4 2- 3 2 - 4 3- 5 3 - 6 3 - 7 4 - 5

Fraction 0.750 0.250 0.25 0.75 0.25 0.50 0.25 0.25

Flow 22.500 7.500 7.50 22.50 7.5 15 7.50 7.50

Arc 4 – 6 4 – 7 5 – 8 5 - 9 6 - 8 6 - 9 7 - 8 7 - 9

Fraction 0.25 0.50 0.250 0.750 0.41 0.59 0.75 0.25

Flow 7.50 15 3.75 11.25 9.37 13.13 16.88 5.63

Operating cost $3,776.25

The first significant point is the fact that the flow between stations is not an integer value, unlike

a conventional minimum cost flow model. In the new solution, unlike Case A, the flow is

distributed throughout the network (Figure 7).

1

2

3

4

5

6

8

9

7


Figure 7. Solution to case B1



The flow between stations requires the sending of material from station 1 to 4 and from station 2

to 3. The same happens with the flow from 3 to 5, 3 to 6, 3 to 7, 4 to 6, 4 to 7, 5 to 8, 6 to 8 and 7

to 9.

The operating cost obtained, being $3,776.25, is higher, which represents a difference of $206.25

or 2.8% in respect of case A. It is worth mentioning that this is normal as, owing to the conditions

imposed on each node and the flow between stations, arcs were employed that had not been

considered, thus raising said cost.

Additional information is obtained with model B1 for the administrators and decision-makers:

We observe, for example, that station 1 is only used at 30% of its used capacity ( or congestion);

stations 2 and 3 are used at 84 and 81% of their capacity, respectively; while stations 6 and 7 are

used at 67.5% of their capacity (Table 8).

Table 8. Properties in each node: analytical vs simulation

Sta. Capacity

(%)

Cycle

time

(days)

Work in

process

(units)

Capacity

(%)

Cycle time

(days)

Work in

process

(units)

Analytical Simulated

1 30.0% 0.014 0.429 30.79% 0.014 0.4375

2 84.0% 0.175 5.250 86.18% 0.239 7.34

3 81.0% 0.142 4.263 81.32% 0.143 4.33

4 90.0% 0.300 9.000 90.81% 0.277 8.44

5 90.0% 0.600 9.000 90.15% 0.539 8.33

6 67.5% 0.092 2.077 66.70% 0.091 2.055

7 67.5% 0.092 2.077 65.89% 0.085 1.935

Property Analytical Simulated Difference

Work in process 32.10 32.91 2.46%

Total Cycle time 0.535 0.5428 1.4%

One relevant factor is the identification of the bottleneck. One accepted definition establishes that

the bottleneck is the station where the most work in process is to be observed (Lawrence and

Buss, 1995). In this example, stations 4 and 5 can both be considered to set the limits for the

production capacity, as they are used at 90% of their capacity.



The cycle time for the process is the total time that elapses from the moment when the production

order is received until it arrives at its destination with the customer: In the example, the total

cycle time is 0.535 days and if we assume a basis of 1 month= 30 days, then the cycle time will

be 16.05 days.

The amount of work (customers or production orders queuing in front of the station waiting for

service) in the overall process is 32.096.

As regards the properties obtained by simulation, the work in process obtained by the analytical

method differs by -2.46% in respect of results of the simulation, while the cycle time differs by-

1.43%.

Case B2

The flows and cost that were found by analytical methods are given in table 9.In this case we

observe that on being compared with case B1, the solution obtained presents several changes in

both the flows and the properties of the system.

Table 9. Flow at each node

Arc 1 - 3 1- 4 2- 3 2 – 4 3- 5 3 - 6 3 - 7 4 - 5

Fraction 0.750 0.250 0 1 0.25 0.50 0.25 0.25

Flow 22.500 7.500 0 30 5.63 11.25 5.63 9.38

Arc 4 - 6 4 - 7 5 - 8 5 – 9 6 - 8 6 - 9 7 - 8 7 - 9

Fraction 0.25 0.50 0.250 0.750 0.386 0.617 0.75 0.25

Flow 9.38 18.75 3.75 11.25 7.97 12.66 18.28 6.09

Operating cost $3744.78

In the flows we observe, for example, that the arc 2 – 3 is not used. The total operating cost,

which in this case is only the flow between pairs of nodes, is less in respect of case B1,

presenting a reduction in cost of 0.84%.

As regards the performance measurements, case B2 presents a bottleneck at stations 4 and 5,

where both are employed at 90% of their capacity, also there is a drop in the work in process and

time cycle (Table 10).



A saving of 5.7% is obtained in the cycle time and there is a reduction of 5.67% in the work in

process with respect to case B1.The analytical results for the work in process and the cycle time

differ by -4.24% and -4.30%, respectively.

Table 10. Optimum service time and properties at each node: analytical vs simulation

Sta.

Optimum

service

time

(days)

Capacity

(%)

Cycle time

(days)

Work in

process

(units)

Capacity

(%)

Cycle

time

(days)

Work in

process

(units)


1 0.01 30.00% 0.014 0.429 29.90% 0.014 0.423

2 0.028 84.00% 0.175 5.25 83.80% 0.186 5.571

3 0.027 60.80% 0.069 1.548 59.80% 0.068 1.507

4 0.024 90.00% 0.24 9 90% 0.244 9.200

5 0.06 90.00% 0.6 9 91% 0.648 9.750

6 0.03 61.90% 0.079 1.623 61.20% 0.080 1.651

7 0.03 73.10% 0.112 2.721 73.28% 0.113 2.773


Work in process 29.57 30.88 4.24%

Total cycle time 0.493 0.5152 4.30%

Cases A, B1 and B2 are comparable with each other. Figure 8 shows the comparison in cost of

case A with cases B1 and B2. The cost of model B2 gives a better solution than case B1, which

means it is worth considering the service time as a variable in order to improve the performance

of the system.

As we have already mentioned, case A gives a lower cost, however this does not imply that it is a

better solution, on the contrary, this is because additional elements are being incorporated into the

model, to with: constraint on the flow, constraints on each node; all the above results in the

feasible region shrinking.



$3,450

$3,500

$3,550

$3,600

$3,650

$3,700

$3,750

$3,800

Case A Case B1 Case B2

Operation Cost ($)

Figure 8. Costs of cases A, B1 and B2

Case B3

This model incorporates the service costs and queuing cost at each station. The values, which are

given in table 11, are arbitrary as the only criterion for assigning them was for the queuing cost to

be more expensive than the service cost. The rest of the model does not have any alterations. The

comparisons shall only be made with model B4, which is described further on.

Table 11. Service cost data and queuing cost

Node 1 2 3 4 5 6 7

Service Cost $5 $3 $16 $10 $4.50 $8 $8

Queueing cost $6.50 $15 $17 $15 $12 $13 $11

Table 12. Flow through each arc

Arc 1 - 3 1- 4 2- 3 2 - 4 3- 5 3 - 6 3 – 7 4 - 5

Fraction 0.750 0.250 0.292 0.708 0.25 0.5 0.25 0.25

Flow 22.500 7.500 8.765 21.235 7.82 15.64 7.82 7.18

Arc 4 - 6 4 - 7 5 – 8 5 - 9 6 - 8 6 - 9 7 – 8 7 - 9

Fraction 0.250 0.50 0.250 0.750 0.421 0.578 0.75 0.25

Flow 7.184 14.36 3.75 11.25 9.62 13.21 16.63 5.54

With the flows given in table 12, the properties of the system are shown below: used capacity,

cycle time and work in process (Table 13).



Table 13. Properties in each node: analytical vs simulation

Sta. Capacity

(%)

Cycle

Time

(days)

Work in

process

(units)

Capacity

(%)

CycleTime

(days) Work in process

(units)


1 30.0% 0.014 0.429 29.43 0.014 0.417

2 84.0% 0.175 5.250 84.52 0.187 5.63

3 84.5% 0.174 5.434 82.97 0.154 4.75

4 86.2% 0.217 6.225 85.64 0.208 5.98

5 90.0% 0.600 9.000 89.14 0.528 7.92

6 68.5% 0.095 2.171 67.32 0.091 2.04

7 66.5% 0.090 1.989 67.32 0.088 1.95


Work in process 30.496 28.712 6.21%


The first significant fact are the overall results obtained for cycle time and work in process, which

are lower than those for case B1. The work in process obtained of 30.496 units is 1.60 units

fewer, which represents a 5 % reduction. In the case of the cycle time, the result is 0.508 days and

represents a difference of 0.027 units; once again if we assume that 1 month =30 days, then the

new cycle time is 15.248 days and represents a 5% improvement in respect of case B1.

Here it is worth mentioning that a desirable goal for any administrator is to shorten the customer's

queuing time and decrease the inventory in process. In both cases the customer would perceive an

improvement in the service they receive: less of a wait to receive their product or else less of a

wait to be served.

We can also see that there is only one bottleneck in this example and this is at station 5, which is

at 90% of its capacity, having on average 9 units in a queue, which is followed, in terms of

holdups, by stations 2, 3 and 4.

The analytical results for the work in process now show a difference of 6.21% and in the case of

the cycle time, of 5.3% in respect of the results obtained by simulation.

As for the costs, having analyzed table 14 in detail, we observe that station 3 represents 26.7% of

the total costs and what stands out is that this has the highest service cost, in other words, service

is expensive at this station. Furthermore, when we look at the queuing cost column, we can see

that station 5 has the highest congestion cost, therefore, from a monetary perspective, queuing at

this station is the most expensive.



Table 15 shows the costs of transport, the operating cost of the stations and the total cost of the

system. In this example in particular, the cost of the transport represents approximately 60% of

the total cost.

Table 14. Analysis of cost per station

Station Service cost WIP cost $ Total / station %

1 $500.00 $2.79 $502.79 19.6%

2 $107.14 $78.75 $185.89 7.2%

3 $592.59 $92.37 $684.97 26.7%

4 $333.33 $93.37 $426.71 16.6%

5 $75.00 $108.00 $183.00 7.1%

6 $266.67 $28.22 $294.88 11.5%

7 $266.67 $21.87 $288.54 11.2%

Table 15. Overall costs

Operating cost of the stations $2566.78 40.44%

Transport cost $3,781.69 59.56%

Total operating cost $6,348.47

Conventionally in a queuing network, an administrator focuses their attention on the bottleneck,

however and as we can see in the results described above, there are other considerations that

should not be ruled out.

In a situation like the one in the example being analyzed, the decision-maker has several possible

directions where they could propose improvements: they can take individual (service, congestion)

costs into account; consider the overall cost; focus on suggesting improvements either to temper

the effect of the production bottleneck or for the cost of transport which, in this case, represents

60% of the system’s total operating cost.

Case B 4.

In this variant we must now determine the service times for stations that minimize the cost of

flow in the network and take the values pij between pairs of nodes as fixed, so there is no change

in the flows. The values are taken from the model B3 solution and Table 16 shows the flows that

are obtained by analytical methods.



Table 16. Flow through each arc

Arc 1 - 3 1- 4 2- 3 2 - 4 3- 5 3 - 6 3 – 7 4 - 5

Fraction 0.750 0.250 0 1 0.25 0.5 0.25 0.5

Flow 22.500 7.500 0 30 5.63 11.25 5.63 18.75

Arc 4 - 6 4 – 7 5 – 8 5 - 9 6 - 8 6 - 9 7 – 8 7 - 9

Fraction 0.250 0.25 0.250 0.750 0.613 0.387 0.75 0.25

Flow 9.38 9.38 6.09 18.28 12.66 7.97 11.25 3.75

The second column in table 17 shows the optimum service time. Looking at the results of case B4

we can see that the solution tends to balance the used capacity of each station: stations 2, 5, 6 and

7, at least, show very little difference between each other.

In this case the bottleneck is to be found at station 4 where the used capacity is 83.33%, followed

by stations 1 and 3.

Table 17. Optimum service time and properties at each node: analytical vs simulation

Sta.

Optimum

service

time

(days)

Capacity

(%)

Cycle

time

(days)

Work in

process

(units)

Capacity

(%)

Cycle

time

(days)

Work in

process

(units)


1 0.028 82.77% 0.160 4.804 82.22% 0.147 4.437

2 0.024 71.01% 0.082 2.449 70.74% 0.086 2.555

3 0.037 82.15% 0.205 4.601 81.97% 0.182 4.045

4 0.022 83.33% 0.133 5.000 83.42% 0.135 5.051

5 0.031 75.14% 0.124 3.023 74.07% 0.118 2.847

6 0.038 78.08% 0.173 3.562 79.23% 0.185 3.892

7 0.051 76.76% 0.220 3.303 76.85% 0.230 3.460


Work in process 26.74 26.38 1.36%


Station 3 contributes with 31.9% of the total costs of the system, followed by station 4 (Table

18).



Table 18. Analysis of cost per station

Sta. service-cost Queuing cost $ Total / station %

1 $ 181.23 $ 31.22 $ 212.45 9.9%

2 $ 126.74 $ 36.74 $ 163.48 7.6%

3 $ 592.76 $ 92.20 $ 684.96 31.9%

4 $ 352.84 $ 65.62 $ 418.47 19.5%

5 $ 95.96 $ 28.46 $ 124.42 5.8%

6 $ 231.28 $ 48.71 $ 279.99 13.0%

7 $ 221.62 $ 44.18 $ 265.80 12.4%

It is worth pointing out that both the transport cost and the operating cost of the stations are

slightly lower than the ones obtained in case B4 (Table 19).

Table 19. Overall costs

Operating cost of

the stations: $2,049.94 36.24%

Transport cost $3,779.53 63.76%

Total operating cost $ 5,829.47

Cases B3 and B4 are compared in Figure 9. We can again see that it is a good idea to treat the

service time as a variable, as this makes it possible to get a better solution: The savings in costs

we obtain through finding the optimum service time and the optimum flow between stations is

6.58%, while the time cycle decreases by 10.83%. The comparison of B3 and B4 with cases A,

B1 and B2 is not considered valid because said models represent other systems as they do not

include either service costs or queuing costs.

$5,500

$5,600

$5,700

$5,800

$5,900

$6,000

$6,100

$6,200

$6,300

$6,400

Case B3 Case B4

Operation Cost ($)

Figure 9. Costs of cases B3 and B4



Conclusions

The incorporation of new elements into a known model enriches the information that can be

extracted on obtaining its solution.

In the case of the networks flow, the conventional minimum cost flow model can be expanded by

assuming that each node is a station that provides a service j has a mean of Ts. In this case the

addition of elements from the queuing networks makes it possible to obtain additional

information about how the resources are being employed as well as, for example, getting metrics

such as the time cycle and quantity of work in process. Equations of queeing systems take into

account the stochastic component of a supply chain.

In this paper we tested a model for a single product and a single server, and can be applied to the

analysis of a supply chain, however it can be broadened to work with a variety of products, in

which case it falls within the multi-commodity flow problem with which a line of action is

decided.

Experiments showed that the optimal solution obtained with the minimum cost flow model

changes if the nodes are considered as queueing systems. Another fact observed is that, when the

nodes are queuing systems, the optimal solution obtained with the model tends to balance the

workload in the nodes and also tends to distribute the flows through the network.

The equations for the properties (WIP and cycle time) are analytic approximations. We observe

in the tests that were done that there is a difference between the analytical and simulated results,

with the biggest difference being observed in case B3 and the smallest difference being observed

in case B4. Even so, we can see that the analytical results have an acceptable level of accuracy.

Under the assumption of Poisson demand, service time exponentianlly distributed and stability of

the system, managers and practitionares can expect that the proposed analytical model can be

considered as a reliable tool for the analysis of a supply chain.

The conventional minimum cost flow model can be expanded to incorporate elements from

queuing networks such as variability of demand, variability of service time, and variability due to

failures or set-ups.

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Appendix 1

The following appendix describes in a general way the sections corresponding to the formulas of

queueing theory coded in the template. It follows the sequence proposed in Curry and Feldman

(2009). The template calculates properties of systems with up to 10 stations of G/G/1 queueing

systems.

Figure A1. Partial view of the spreadsheet template: Paths, properties of the network, external

flows and costs of the queues

The template is divided into the follows sections: Process path (probability matrix), External

flows, Properties of the network (figure A1); also with the costs of the stations, Table of origin-

destination costs, flow between nodes, cost of the flow between nodes, Operating cost of the

stations, transport cost and total cost of the system (figure A2).

The shaded cells are the data that should be entered, for example: External flows (C16:C25) and

their standard deviation of the arrivals (D16:D25), service times (O3:O12) with their standard

deviation (P3:P12). Other data are service costs (O16:O25), queuing costs (P16:P25) and the

costs of transfer between nodes(X3:AG12).



Figure A2. Partial view of the spreadsheet template: Costs of transport and flows

Flow in the network

First it is necessary evaluate the arrival flow to each station with eq. (6). Once the user has

entered all data, the spreadsheet evaluates the transpose with the following function:

=TRANSPOSE (C3:L12)

After, the spreadsheet calculates the difference Pt- I and then gets its inverse:

=MINVERSE(C63:L72)

The arrivals to each station are obtained by multiplying the vector of input flows by the inverse:

=MMULT(C75:L84;C16:C25)

Variability of the arrivals

To solve eq. (8) the procedure is as follows:

Each element of Q is calculated with eq. (9). For example:

=IF ($G$16>0;G16*C3*C3*(AC34)/$G$16; )

The template transposes Q:

=TRANSPOSE(P34:Y43)

Evaluates (I-Q) and then obtains the inverse:

=MINVERSE(P84:Y93)

Each element of β vector is obtained in the range (P47:P53), (R47:AA56) and (AD47:AD53)

using eq. (10). In the spreadsheet the formulas are:



IF(G16>0;(C16/G16)*F16;) + IF(G16>0;($G$16*C33/G16)*(C33*$AB$34*$Q$3+1-C33);) +

…+ IFI(G16>0;($G$25*L33/G16)*(L33*$AB$43*$Q$12+1-L33); )

Function IF is used to avoid division by zero.

In the final step Ca2 is obtained with Eq. (8), in the spreadsheet the formula is:

=MMULT(P95:Y104;AE47:AE56)

Cycle time and Work in Process

The template updates automatically the cells. The cycle time (T3:T12) and work-in-process

(U3:U12) are obtained applying equations (11), (12) and (22).

Documents

Revista Electrónica Nova ScientiaHernández-González, S. et al. Revista Electrónica Nova Scientia Numerical analysis of minimum cost network flow with queuing stations: the M/M/1